# Questions tagged [cauchy-schwarz-inequality]

The Cauchy-Schwarz inequality states $|\langle x,y \rangle |\leq ||x||\cdot ||y||.$ Use this tag for questions related to the CS inequality and its applications.

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### Prove that sum of eigenvalues of the inverse of an nxn correlation matrix A is greater than or equal to n

I stuck on this question and here is my thoughts: So we have a nxn correlation matrix A with eigenvalues: λ_1,λ_2,...,λ_n 1.According to the property of correlation matrix, (λ_1)+(λ_2) + ... + (λ_n) = ...
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### How to prove that $1/ ((y+z) x^4) + 1/ ((z+x) y^4) + 1/ ((x+y) z^4) \geq 3/2$ for $x, y, z>0$ such that $xyz=1$? [closed]

How to prove that $\dfrac{1}{(y+z) x^4} + \dfrac{1}{(x+z) y^4} + \dfrac{1}{(y+x) z^4}\geq3/2$ for $x, y, z>0$, such that $xyz=1$?
1 vote
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### The inequality $\int^\infty_0 (\sin(rt)r^3/\sinh^2(r)) dr\leq cte^{-At}$

How to prove the following inequality $$\forall t>0,\quad\int^\infty_0 \sin(rt)\frac{r^3}{\sinh^2(r)} dr\leq c \big(te^{-At}\big)$$ for some constants $A>0,c>0$
1 vote
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### Are these $L_2$-spectral radii approximations strictly increasing?

Suppose that $V$ is a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear mappings from $V$ to $V$. Let $A_1,\dots,A_r:V\rightarrow V$ be linear operators. Then ...
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### When does the Cauchy-Schwarz inequality for spectral radii of tensor products become equality?

Let $V$ be a complex finite dimensional inner product space. If $A_{1},\dots,A_{n}:V\rightarrow V$ are linear operators, then let $\Phi(A_{1},\dots,A_{n}):L(V)\rightarrow L(V)$ be the superoperator ...
1 vote
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### General bivariate functions that satisfy Cauchy-Schwarz

Have there been any study of general bivariate functions $f:X \times X \to \mathbb{R}$ that satisfy $f(x,y)^2 \leq f(x,x)f(y,y)$. This comes up as a function I'm working with satisfies the asymmetric ...
1 vote
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### Understanding statement about bounds of vector in the context of a RSDF ≤ₘ WOPT proof

I'm trying to follow the proof of Lemma 4 of "Strong NP-Hardness of the Quantum Separability Problem", by S. Gharibian, 2010 , which, roughly, states that there is a many-one reduction ...
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### Properties of a function $C_\ell(\ell)$ which checks an inequality in ideal case (decreasing assumption) and after estimating impact in general case

Suppose that $X=2 \ell+1, Y=C_{\ell}$, both $X$ and $Y$ are function of $\ell$, $X$ is increasing and $Y$ is assuming to be decreasing. But in reality, my data follow a $C_\ell$ increasing for a small ...
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Let $E$ be a (right) Hilbert module over the $C^*$-algebra $B$. Let $\phi$ be a state on the $C^*$-algebra $B$. Then consider $$N_\phi:= \{x \in E: \phi(\langle x,x\rangle)=0\}.$$ I want to show that $... 4 votes 2 answers 401 views ### A completely positive equivariant map$\varphi: A \to B$induces a map$A \rtimes_r \Gamma \to B \rtimes_r \Gamma.$Recall the construction of the reduced crossed product: Let$\Gamma$be a discrete group and$A$be a$C^*$-algebra with an action$\alpha: \Gamma\to \operatorname{Aut}(A)$. Consider the$*$-algebra$... 1 vote
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### Did Euler ever use anything similar to Cauchy's inequality?

This could be asked more provocatively, indeed how it arose, as "how did Euler do so much mathematics without using and/or knowing Cauchy's inequality?", something that came up in the ...
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### Does the Cauchy–Schwarz inequality imply 2-positivity?

Recall the following generalisation of Cauchy–Schwarz. Theorem. Let $f\colon \mathscr{A} \to \mathscr{B}$ be a linear 2-positive map between C$^*$-algebras. Then for all $a,b \in \mathscr{A}$ we ...
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